SL(3,3^n) 和 SU(3,3^n)的第一 Cartan 不变量*

确定Cartan不变量是代数群与相关的李型有限群的模表示理论中的一个重要方面.作者利用代数群模表示理论中的一系列结果,计算了3~n个元素的有限域上特殊线性群SL(3,3~n)和特殊酉群SU(3,3~n)的第一Cartan不变量,得到如下结论:当G=SL(3,3~n)时,C_(00)~((n))=a~n+b~n+6~n-2·8~n;而当G=SU(3,3~n)时,C_(00)~((n))=a~n+b~n+6~n-2·8~n+2·(1+(-1)~n),其中a,b是多项式x~2-20x+48的两个根.另外,作者也得到了射影不可分解模U_n(0,0)的维数公式:dim U_n(0,0)=(12~n-6~n+∈)·3~(3n),其中,当G=SL(3,3~n)时,∈=1;而当G=SU(3,3~n)时,∈=-1.     

Order of elements in the groups related to the general linear group

For a natural number n and a prime power q the general, special, projective general and projective special linear groups are denoted by GL_n(q), SL_n(q), PGL_n(q) and PSL_n(q), respectively. Using conjugacy classes of elements in GL_n(q) in terms of irreducible polynomials over the finite field GF(q) we demonstrate how the set of order elements in GL_n(q) can be obtained. This will help to find the order of elements in the groups SL_n(q), PGL_n(q) and PSL_n(q). We also show an upper bound for the order of elements in SL_n(q).     

A presentation for reduced extended affine weyl groups

In 1985 K. Saito [Sal] introduced the concept of an extended affine Weyl group (EAWG), the Weyl group of an extended affine root system (EARS). In [A2, Section 5J, we gave a presentation called “a presentation by conjugation” for the class of EAWGs of index zero, a subclass of EAWGs. In this paper we will give a presentation wh.ich we call a “generalized present.ation by conjugation” for the class of reduced EAWGs. If the extended affine Weyl group is of index zero this presentation reduces to “a presentation by conjugation”. Our main result states that when the nullity of the EARS is 2, these two presentations coincide that is, EAWGs of nullity 2 have “a presentation by conjugation”. In [ST] another presentation for EAWGs of nullity 2 is given.     

保持粘切的加法满射及其应用

设m 2,Mm(D)是除环D上的所有m阶矩阵构成的加法群.本文应用矩阵几何基本定理刻划了Mm(D)上保持粘切的加法满射.作为应用,还给出了Mm(D)上保持一般线性群的加法满射的形式.     

关于四个三幂等阵线性组合的可逆性和群逆

本文研究了四个三幂等阵线性组合的可逆性及群逆.利用矩阵分解的方法,获得了它们可逆及群逆的一些条件,并得到其逆和群逆的计算公式,这些结论完善了k幂等阵可逆性理论.     

Compound matrices: properties,numerical issues and analytical computations

This paper studies the possibility to calculate efficiently compounds of real matrices which have a special form or structure. The usefulness of such an effort lies in the fact that the computation of compound matrices, which is generally noneffective due to its high complexity, is encountered in several applications. A new approach for computing the Singular Value Decompositions (SVD’s) of the compounds of a matrix is proposed by establishing the equality (up to a permutation) between the compounds of the SVD of a matrix and the SVD’s of the compounds of the matrix. The superiority of the new idea over the standard method is demonstrated. Similar approaches with some limitations can be adopted for other matrix factorizations, too. Furthermore, formulas for the n − 1 compounds of Hadamard matrices are derived, which dodge the strenuous computations of the respective numerous large determinants. Finally, a combinatorial counting technique for finding the compounds of diagonal matrices is illustrated.      

On a minimal set of generators for the invariants of 3 × 3 matrices

Let K be a field of characteristic p > 0, let L be a restricted Lie algebra and let R be an associative K-algebra. It is shown that the various constructions in the literature of crossed product of R with u(L) are equivalent. We calculate explicit formulae relating the parameters involved and obtain a formula which hints at a noncommutative version of the Bell polynomials.     

Symmetric Weighing Matrices Constructed using Group Matrices

A weighing matrix of order n and weight m² is a square matrix M of order n with entries from {-1,0,+1} such that MM^T=m²I where I is the identity matrix of order n. If M is a group matrix constructed using a group of order n, M is called a group weighing matrix. Recently, group weighing matrices were studied intensively, especially when the groups are cyclic and abelian. In this paper, we study the abelian group weighing matrices that are symmetric, i.e.M^T=M. Some new examples are found. Also we obtain a few exponent bounds on abelian groups that admit symmetric group weighing matrices. In particular, we prove that there is no symmetric abelian group weighing matrices of order 2p^r and weight p² where p is a prime and p≥ 5.Communicated by: K.T. Arasu     

On Non-Congruence Subgroups of the Analogue of the Modular Group in Characteristic p

Let k[t] be the polynomial ring over a finite field k. The group SL ₂(k[t]) is often referred to as the analogue, in characteristic p, of the classical modular group SL ₂( ), where is the ring of rational integers. It is well-known that the smallest index of a non-congruence subgroup of SL ₂( ) is 7. Here we compute this index for SL ₂(k[t]). (In all but 6 cases it turns out to be 1 + q, where q is the order of k.)     

Fischer Matrices of the Affine Groups

Let GL_n(q) be the general linear group and let H_n ; V_n(q) · GL_n(q) denote the affine group of V_n(q). In [1] and [4], we determined Fischer matrices for the conjugacy classes of GL_n(q) where n = 2, 3, 4 and we obtained the number of conjugacy classes and irreducible characters of H₂, H₃, and H₄. In this paper, we find the Fischer matrices of the affine group H_n for arbitrary n.AMS Subject Classification Primary 20C15 Secondary 20C33